Curve sketching: increasing and decreasing functions

When sketching the graph of a function, among other things we work out the intervals on which the function is increasing or decreasing. The derivative of the function helps us a great deal here.

How to find where a function is increasing or decreasing

The slope (gradient) of the tangent at a point, that is, the derivative of the function at that point, tells us more about the function than you might expect. First, take a look at the following diagram:

Here is what we can notice – if the function is decreasing at a given point, then the tangent at that point is decreasing too (more precisely, the function that would describe that tangent is decreasing). You can see this with tangent b. The function x² + 1 is decreasing at the point B, and the tangent b is decreasing as well. Tangent a, on the other hand, is increasing, and we can see that the function x² + 1 is also increasing at the point A. The third tangent c is parallel to the x-axis, and we can see that there is a maximum or minimum at this point – specifically a minimum.

You can see these things at a glance, but we need to make them more precise. We start with increasing and decreasing functions. If the tangent to a function at a point is increasing, then the function is increasing at that point too. When is a tangent increasing? A tangent is increasing if the angle it makes with the positive x-axis is in the interval (0, 90) degrees. A tangent is decreasing if the angle it makes with the positive x-axis is in the interval (90, 180) degrees. Take a look at the following diagram:

The angle α is definitely less than 90 degrees, and the tangent is increasing. The angle β is greater than 90 degrees, and the tangent is decreasing. That should be clear. Now back to the tangent function. We need to recall how tan is calculated, and for that we need the unit circle. We mark the angles we want on the unit circle, and the unit circle gives us the values of the trigonometric functions according to certain rules. The value of tan is shown as the y-coordinate of the point where the arm of the angle meets the perpendicular to the x-axis drawn at the point (1, 0). The diagram follows:

In the diagram we have the blue unit circle and two lines, the red line a and the green line b. These lines make angles α and β with the positive x-axis. We can easily read off the tan of these angles from the diagram: we draw the perpendicular p to the x-axis at the point (1, 0). That is the grey line. The tan of the angle α is then equal to the y-coordinate of the point where the red line a meets the grey line p. In other words, the tan of the angle α is equal to the length of the line segment |BP|.

The same goes for the line b and the angle β. The tan of the angle β is equal to the y-coordinate of the point where the line b meets the line p. That is, the negative of the length of the line segment CP.

As you can see, tan(α) is positive, whereas tan(β) is negative. And that is what we were looking for. As the diagram shows, if the angle is in the interval (0, 90), as the angle α is, then its tan is positive, and if it is in the interval (90, 180), as the angle β is, then it is negative.

How can we now easily find the angle that the tangent at a given point makes with the positive x-axis? By using derivatives. The derivative of a function gives us the slope of the tangent at a given point, and the slope of the tangent is precisely the tan of the angle the tangent makes with the positive x-axis. In other words, if the first derivative of the function is positive at a given point, then the function is increasing there. If it is negative, then it is decreasing.

If f′(x) > 0, then the function f is increasing at the point x. If f′(x) < 0, then the function f is decreasing at the point x.

Example: let's use the function f(x) = x² again. We know that its derivative is f′(x) = 2x. The function is decreasing at all points where 2x < 0, and increasing at all points where 2x > 0. These are simple inequalities, so we get the intervals (−∞, 0) for decreasing and (0, ∞) for increasing.